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| In [[mathematics]], a '''split-biquaternion''' is a [[hypercomplex number]] of the form
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| :<math>q = w + xi + yj + zk \!</math>
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| where ''w'', ''x'', ''y'', and ''z'' are [[split-complex number]]s and i, j, and k multiply as in the [[quaternion group]]. Since each [[coefficient]] ''w'', ''x'', ''y'', ''z'' spans two [[real number|real]] [[dimension]]s, the split-biquaternion is an element of an eight-dimensional [[vector space]]. Considering that it carries a multiplication, this vector space is an [[algebra over a field|algebra]] over the real field, or an [[algebra over a ring]] where the split-complex numbers form the ring. This algebra was introduced by [[William Kingdon Clifford]] in an 1873 article for the [[London Mathematical Society]]. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the [[tensor product of algebras]], and as an illustration of the [[direct sum of modules#Direct sum of algebras|direct sum of algebras]].
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| The split-biquaternions have been identified in various ways by algebraists; see the ''Synonyms'' section below.
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| ==Modern denomination==
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| A split-biquaternion is a member of the [[Clifford algebra]] ''C''ℓ<sub>0,3</sub>('''R'''). This is the [[geometric algebra]] generated by three orthogonal imaginary unit basis directions, {''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>} under the combination rule
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| ::<math>e_i e_j = \Bigg\{ \begin{matrix} -1 & i=j, \\
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| - e_j e_i & i \not = j \end{matrix} </math>
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| giving an algebra spanned by the 8 basis elements {1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>, ''e''<sub>2</sub>''e''<sub>3</sub>, ''e''<sub>3</sub>''e''<sub>1</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>}, with (''e''<sub>1</sub>''e''<sub>2</sub>)<sup>2</sup> = (''e''<sub>2</sub>''e''<sub>3</sub>)<sup>2</sup> = (''e''<sub>3</sub>''e''<sub>1</sub>)<sup>2</sup> = −1 and (ω = ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>)<sup>2</sup> = +1.
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| The sub-algebra spanned by the 4 elements {1, ''i'' = ''e''<sub>1</sub>, ''j'' = ''e''<sub>2</sub>, ''k'' = ''e''<sub>1</sub>''e''<sub>2</sub>} is the [[division ring]] of Hamilton's [[quaternions]], '''H''' = ''C''ℓ<sub>0,2</sub>('''R''')
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| One can therefore see that
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| :<math>C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \otimes \mathbb{D}</math>
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| where '''D''' = ''C''ℓ<sub>1,0</sub>('''R''') is the algebra spanned by {1, ω}, the algebra of the [[split-complex number]]s.
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| Equivalently,
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| :<math>C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \oplus \mathbb{H}.</math>
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| <!--
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| The idea is to replace the [[complex number]]s used in an ordinary (Hamilton) [[biquaternion]] with [[split-complex number]]s.Thus q = w + x i + y j + z k , with w, x, y, z ∈ '''D''' is a Clifford biquaternion. Such a number can also be written q = r + s ω , r, s ∈ '''H''', ω<sup>2</sup> = + 1 , '''H''' the [[division ring]] of Hamilton's [[quaternions]]. -->
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| ==Split-biquaternion group==
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| The split-biquaternions form an [[associativity|associative]] [[ring theory|ring]] as is clear from considering multiplications in its [[basis (linear algebra)|basis]] {1, ω, i, j, k, ωi, ωj, ωk,}. When ω is adjoined to the [[quaternion group]] one obtains a 16 element group
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| :( {1, i, j, k, −1, −i, −j, −k, ω, ωi, ωj, ωk, −ω, −ωi, −ωj, −ωk}, × ).
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| ==Direct sum of two quaternion rings==
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| The direct sum of the division ring of quaternions with itself is denoted <math>\mathbb{H} \oplus \mathbb{H}</math>. The product of two elements <math>(a \oplus b)</math> and <math> (c \oplus d)</math> is <math> a c \oplus b d </math> in this [[direct sum of modules#Direct sum of algebras|direct sum algebra]].
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| '''Proposition:''' The algebra of split-biquaternions is isomorphic to <math>\mathbb{H} \oplus \mathbb{H}.</math>
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| proof: Every split-biquaternion has an expression ''q'' = ''w'' + ''z'' ω where ''w'' and ''z'' are quaternions and ω<sup>2</sup> = +1. Now if ''p'' = ''u'' + ''v'' ω is another split-biquaternion, their product is
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| :<math> pq = uw + vz + (uz + vw) \omega .\!</math>
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| The isomorphism mapping from split-biquaternions to <math>\mathbb{H} \oplus \mathbb{H}</math> is given by
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| :<math>p \mapsto (u + v) \oplus (u - v) , \quad q \mapsto (w + z) \oplus (w - z).</math> | |
| In <math>\mathbb{H} \oplus \mathbb{H}</math>, the product of these images, according to the algebra-product of <math>\mathbb{H} \oplus \mathbb{H}</math> indicated above, is
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| :<math>(u + v)(w + z) \oplus (u - v)(w - z).</math>
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| This element is also the image of pq under the mapping into <math>\mathbb{H} \oplus \mathbb{H}.</math>
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| Thus the products agree, the mapping is a homomorphism; and since it is [[bijective]], it is an isomorphism.
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| Though split-biquaternions form an [[eight-dimensional space]] like Hamilton’s biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions.
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| == Hamilton biquaternion ==
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| The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by [[William Rowan Hamilton]]. Hamilton's [[biquaternion]]s are elements of the algebra
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| :<math>C\ell_2(\mathbb{C}) = \mathbb{H} \otimes \mathbb{C}.</math>
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| ==Synonyms==
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| The following terms and compounds refer to the split-biquaternion algebra:
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| * elliptic biquaternions – Clifford (1873), Rooney(2007)
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| * octonions – [[Alexander MacAulay]] (1898)
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| * Clifford biquaternion – Joly (1902), van der Waerden (1985)
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| * dyquaternions – Rosenfeld (1997)
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| * <math>\mathbb{D} \otimes \mathbb{H}</math> where '''D''' = [[split-complex number]]s – Bourbaki (1994), Rosenfeld (1997)
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| * <math>\mathbb{H} \oplus \mathbb{H}</math>, the [[direct sum of modules#Direct sum of algebras|direct sum]] of two quaternion algebras – van der Waerden (1985)
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| ==See also==
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| * [[split-octonion]]s
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| ==References==
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| *[[William Kingdon Clifford]] (1873), "Preliminary Sketch of Biquaternions", Paper XX, ''Mathematical Papers'', p. 381.
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| *[[Alexander MacAulay]] (1898) ''Octonions: A Development of Clifford's Biquaternions'', Cambridge University Press.
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| *P.R. Girard (1984), "The quaternion group and modern physics", ''European Journal of Physics'', '''5''':25-32.
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| *Joe Rooney (2007) "William Kingdon Clifford", in Marco Ceccarelli, ''Distinguished figures in mechanism and machine science'', Springer.
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| *[[Charles Jasper Joly]] (1905) ''Manual of Quaternions'', page 21, MacMillan & Co.
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| *Boris Rosenfeld (1997) ''Geometry of Lie Groups'', page 48, [[Kluwer]] ISBN 0-7923-4390-5 .
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| *[[Nicolas Bourbaki]] (1994) ''Elements of the History of Mathematics'', J. Meldrum translator, Springer.
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| *[[B. L. van der Waerden]] (1985) ''A History of Algebra'', page 188, Springer, ISBN 0-387-13610-X .
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| [[Category:Quaternions]]
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| [[Category:Clifford algebras]]
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| [[de:Biquaternion#Clifford Biquaternion]]
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Greetings! I am Marvella and I really feel comfy when people use the full name. North Dakota is her beginning place but she will have to move one working day or another. What I love doing is to gather badges but I've been using on new issues recently. Bookkeeping is what I do.
Look into my site: at home std test